SECTION 14.2 Algebraic Techniques for Finding Limits 949 Finding the Limit of a Monomial Find: ( ) − → x lim 4 x 2 3 Solution EXAMPLE 6 ( ) − = − ⋅ = − ⋅ = − → x lim 4 4 2 4 8 32 x 2 3 3 Proof If P is a polynomial function—that is, if ( ) = + + + + − − P x a x a x a x a n n n n 1 1 1 0 then [ ] ( ) ( ) ( ) ( ) ( ) = + + + + = + + + + = + + + + = → → − − → → − − → → − − P x a x a x a x a a x a x a x a a c a c a c a P c lim lim lim lim lim lim x c x c n n n n x c n n x c n n x c x c n n n n 1 1 1 0 1 1 1 0 1 1 1 0 ■ Finding the Limit of a Polynomial Find: [ ] − + + − → x x x x lim 5 6 3 4 2 x 2 4 3 2 Solution EXAMPLE 7 [ ] − + + − = ⋅ − ⋅ + ⋅ + ⋅ − = ⋅ − ⋅ + ⋅ + − = − + + = → x x x x lim 5 6 3 42526232422 5 16 6 8 3 4 8 2 80 48 12 6 50 x 2 4 3 2 4 3 2 In Words To find the limit of a polynomial as x approaches c , evaluate the polynomial at c . 2 Find the Limit of a Polynomial Since a polynomial is a sum of monomials, we use the Limit of a Monomial and the Limit of a Sum repeatedly to prove the following theorem: THEOREM Limit of a Polynomial If P is a polynomial function, then ( ) ( ) = → P x P c lim x c for any number c . Now Work PROBLEM 17 3 Find the Limit of a Power or a Root THEOREM Limit of a Power or a Root If ( ) → f x lim x c exists and if ≥ n 2 is an integer, then [ ] ( ) ( ) = ⎡ ⎣⎢ ⎤ ⎦⎥ → → f x f x lim lim x c n x c n and ( ) ( ) = → → f x f x lim lim x c n x c n provided ( ) > f x 0 if n is even. Look carefully at the Limit of a Power or a Root properties and compare each side.
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